PUBLIC SCHOOLS OF EDISON TOWNSHIP OFFICE OF CURRICULUM AND INSTRUCTION GEOMETRY HONORS. Middle School and High School


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1 PUBLIC SCHOOLS OF EDISON TOWNSHIP OFFICE OF CURRICULUM AND INSTRUCTION GEOMETRY HONORS Length of Course: Elective/Required: Schools: Term Required Middle School and High School Eligibility: Grades 812 Credit Value: (High School Only) 5 Credits Date Approved: August 25, 2014
2 GEOMETRY HONORS TABLE OF CONTENTS Statement of Purpose 3 Course Objectives 4 Suggested Time Line (Level: Honors) 7 Unit 0: Chapter 0 Preparing for Geometry 8 Unit 1: Chapter 1  Tools of Geometry 10 Unit 2: Chapter 2  Reasoning and Proof 12 Unit 3: Chapter 3  Parallel and Perpendicular Lines 15 Unit 4: Chapter 4  Congruent Triangles 17 Unit 5: Chapter 5  Relationships in Triangles 20 Unit 6: Chapter 6 Quadrilaterals 22 Unit 7: Chapter 7  Proportions and Similarity 24 Unit 8: Chapter 8  Right Triangles/ Trigonometry 26 Unit 9: Chapter 9  Transformations and Symmetry 28 Unit 10: Chapter 10 Circles 31 Unit 11: Chapter 11 Area of Polygons and Circles 34 Unit 12: Chapter 12 Extending Surface Area/ Volume 36 Unit 13: Chapter 13 Probability and Measurement 38
3 GEOMETRY HONORS 3 Statement of Purpose This course of study has been designed for the Geometry Honors course. The curriculum introduces geometric topics, skills, and concepts. The material should be connected to realworld situations as often as possible, as suggested in the curriculum. This course curriculum guide provides a thorough preparation for geometric questions that may be included on the state assessment. In order to promote the effective implementation of this program, the following suggestions are provided: 1. Formative assessment should be used throughout this course, as with any math course, in order to monitor students learning and instruction such be adjusted as needed. 2. Instruction should be differentiated in order to accommodate the different ways students learn. 3. Students should be encouraged to maintain an organized and thorough set of notes in a notebook. Teachers should indicate the expected format and content and should explain how notebooks can be effectively utilized. 4. Meaningful and relevant homework assignments should be given to students on a regular basis to encourage practice of new skills and concepts. 5. Examples of application of mathematics and specifically Algebra in careers and everydaylife situations should be provided as motivation wherever possible. 6. Students should be required to use correct mathematical terminology at all times. 7. The use of technology is encouraged wherever possible in order to foster the impact on students learning and understanding. 8. Modifications and accommodations should be included where necessary to meet student s Individual Education Plans (IEP).
4 GEOMETRY HONORS 4 The student will be able to: Chapter 1 Course Objectives Develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. Use construction to explore attributes of geometric figures and to make conjectures about geometric relationships. Use one and twodimensional coordinate systems to represent points, lines, rays, line segments, and figures. Find areas of regular polygons, circles, and composite figures. Chapter 2 Use inductive reasoning to formulate a conjecture. Use logical reasoning to prove statements are true and find counterexamples to disprove statements that are false. Determine the validity of a conditional statement, its converse, inverse, and contrapostive. Use deductive reasoning to prove a statement. Chapter 3 Make conjectures about lines and determine the validity of the conjectures. Make conjectures about angles and determine the validity of the conjectures. Use slopes of equations of lines to investigate geometric relationships, including parallel lines and perpendicular lines. Use one and twodimensional coordinate systems to represent lines. Chapter 4 Make conjectures about polygons. Use numeric and geometric patterns to make generalizations about geometric properties. Use logical reasoning to prove statements are true. Chapter 5 Use slope and equations of lines to investigate geometric relationships, including special segments of triangles. Recognize and know historical development of geometric systems and know that mathematics was developed for a variety of purposes. Analyze geometric relationships in order to verify conjectures.
5 GEOMETRY HONORS 5 Chapter 6 Use geometric concepts and properties to solve problems in fields such as art and architecture. Identify and apply mathematics to everyday experience, to activities in and outside of school, with other disciplines, and with other mathematical topics. Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models. Chapter 7 Use ratios to solve problems involving similar figures. Formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models. Chapter 8 Use and extend similarity properties to explore and justify conjectures about geometric figures. Derive, extend, and use the Pythagorean Theorem. Identify and apply patterns from right triangles to solve meaningful problems, including special right triangles ( and ) and triangles with sides that are Pythagorean triples. Develop, apply, and justify triangle similarity relationships, such as trigonometric ratios using a variety of methods. Chapter 9 Use congruence transformations to make conjectures and justify properties of geometric figures. Chapter 10 Find areas of sectors and arc lengths of circles using proportional reasoning. Use numeric and geometric patterns to make generalizations about geometric properties including properties of angle relationships in circles. Chapter 11 Find areas of regular polygons, circles, and composite figures. Find areas of sectors and arc lengths of circles using proportional reasoning.
6 GEOMETRY HONORS 6 Chapter 12 Find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures. Describe the effect on area and volume when one or more dimensions of a figure are changed. Chapter 13 Understand sample spaces and design simulations Compute probabilities for independent, dependent, mutually exclusive, not mutually exclusive, and conditional events. Calculate geometric probabilities. Note  All objectives will include applications to reallife situations. For Common Core Standards in full visit Standards can be implemented in addition to the standards listed in this curriculum.
7 GEOMETRY HONORS 7 Suggested Time Line (Level: Honors) UNIT # of PERIODS COMPLETE FOR EYA Unit 0: Chapter 0 Preparing for Geometry Optional 5 Unit 1: Chapter 1  Tools of Geometry , 1.7 Unit 2: Chapter 2  Reasoning and Proof 12 Unit 3: Chapter 3  Parallel and Perpendicular Lines Unit 4: Chapter 4  Congruent Triangles 18 (end MP 1) 4.1, Unit 5: Chapter 5  Relationships in Triangles Unit 6: Chapter 6  Quadrilaterals Unit 7: Chapter 7  Proportions and Similarity 14 (end MP 2) Unit 8: Chapter 8  Right Triangles/ Trigonometry , 8.7 Unit 9: Chapter 9  Transformations and Symmetry Unit 10: Chapter 10  Circles 14 (end MP 3) , 10.8 Unit 11: Chapter 11 Area of Polygons and Circles , 11.3 Unit 12: Chapter 12 Extending Surface Area/ Volume , 12.4, 12.5, 12.6 Unit 13: Chapter 13 Probability and Measurement 10 (end MP 4) Note  The above suggested time line is a rough guideline based on suggestions from the Glencoe Geometry textbook. Teachers must adjust their timing and pacing as they feel necessary to accommodate actual class periods available.
8 GEOMETRY HONORS 8 Targeted Standards: Unit Title: Chapter 0 Preparing For Geometry Unit Objectives/Conceptual Understandings: The concepts presented in Chapter 0 are review from previous courses. You may wish to use all or some of the chapter at the beginning of the school year to refresh students' skills. Or you may wish to begin with Chapter 1 and use the Chapter 0 lessons as needed to reinforce prerequisite skills as you progress through the program. Unit : Teachergenerated assessments will be used. Progress Indicators CC.912.S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) CC.912.A.CED.4* Students extend their work with algebraic properties and solving equations in one variable to solving literal equations for a given variable. CC.912.A.REI.3 CC.912.A.CED.1* Students have written and solved linear equations in one variable. Now they extend this work to inequalities. The methods for solving inequalities are very similar to the methods for solving equations; many students strengthen their understanding by studying both. As with equations, students will create and use inequalities to solve Unit vocabulary including: a. experiment b. trial c. outcome d. event e. probability f. theoretical probability g. experimental probability h. ordered pair i. xcoordinate j. ycoordinate k. quadrant l. origin m. system of equations n. substitution o. elimination p. Product Property q. Quotient Property Convert units of measure within the customary and metric systems. Convert units of measure between the customary and metric systems. Find the probability of simple events. Use the order of operations to evaluate algebraic expressions. Use algebra to solve linear equations. Use algebra to solve linear inequalities. Name and graph points in Use pretest and posttest provided in the text
9 GEOMETRY HONORS 9 Unit Title: Chapter 0 Preparing For Geometry (cont.) Progress Indicators problems in contextual situations. the coordinate plane. CC.912.A.CED.2* CC.912.A.REI.6 Students have seen that the relationship between two or more variables can be represented as an equation or inequality and as a graph. Understanding equivalent representations and fluently translating between them are important for solving problems. For example, the connection between an equation and its graph is key to understanding how to use a graph to solve a system of equations. Use graphing, substitution, and elimination to solve systems of linear equations. Evaluate square roots and simplify radical expressions. Resources: Essential Materials, Supplementary Materials, Links to Best Practices, Leveled Worksheets, Personal Tutor, Virtual Manipulatives, 5Minute Checks, Graphing Calculator Instructional Adjustments: Leveled Worksheets, Differentiation Resources, Scaffolding Questions
10 GEOMETRY HONORS 10 Unit Title: Chapter 1 Tools of Geometry Targeted Standards: G.CO Experiment with transformations in the plane. G.GMD Explain volume formulas and use them to solve problems. G.MG Apply geometric concepts in modeling situations. G.GPE Translate between the geometric description and the equation for a conic section. Unit Objectives/Conceptual Understandings: Students will be able to: Develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning and theorems. Use construction to explore attributes of geometric figures to make conjectures about geometric relationships. Use one and two dimensional coordinate systems to represent points, lines, rays, line segments, and figures. Find areas of regular polygons, circles, and composite figures. Essential Questions: Why do we measure? Unit : Teachergenerated assessments will be used. G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). G.MG.1 Use geometric Unit vocabulary including: a. collinear b. coplanar c. congruent d. midpoint e. segment bisector f. angle g. vertex h. angle bisector i. perpendicular j. polygon k. perimeter l. volume Identify and model points, lines, and planes. Identify intersecting lines and planes. Identify and model points, lines, and planes. Identify intersecting lines and planes. Measure segments. Calculate with measures. Find the distance between two points. Find the midpoint of a segment. Challenge students to develop threedimensional models to demonstrate difficult geometric concepts related to points, lines, and planes. Some examples include: Develop a model to show that three points can be noncollinear. Develop a demonstration to show that three points are coplanar, but four points can be noncoplanar. Develop a threedimensional model of lines that are not Use formative assessments suggested in the textbook. Use Guided Practice, Check Your Understanding, HOT problems, and Spiral Reviews as needed. Use Ticket Out the Door Section 1.2 and 1.6. Use MidChapter
11 GEOMETRY HONORS 11 Unit Title: Chapter 1 Tools of Geometry (cont.) shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Visualize relationships between twodimensional and threedimensional objects. Measure segments. Calculate with measures. Find the distance between two points. Find the midpoint of a segment. Measure and classify angles. Identify and use congruent angles and the bisector of an angle. Identify and use special pairs of angles. Identify perpendicular lines. Identify and name polygons. Find perimeter or circumference and area of twodimensional figures. Identify and name threedimensional figures. Find surface area and volume. Measure and classify angles. Identify and use congruent angles and the bisector of an angle. Identify and use special pairs of angles. Identify perpendicular lines. Identify and name polygons. Find perimeter, circumference, and area of twodimensional figures. Identify and name threedimensional figures. Find surface area and volume. Resources: Leveled Worksheets, Personal Tutor, Virtual Manipulatives, 5Minute Checks found on connect.ed.mcgrawhill.com Geometer Sketchpad Graphing Calculator Teaching Geometry with Manipulatives parallel and do not intersect. Have students sketch three different segments that each have (0, 0) as a midpoint. Write the coordinates of the endpoints of each segment. What do you notice about the coordinates? Have students list each angle relationship presented in the Key Concept boxes of this lesson on pages Then, have students write one or two sentences to describe each relationship and provide an example. When discussing surface area, provide students with the opportunity to make nets, cut them out, and then put them together to make a solid. This should help them better understand surface area. Instructional Adjustments: Quiz Lesson p 45. Use SelfCheck Quizzes as needed. Use Leveled Worksheets. Use Lesson Resources to provide differentiation. Use scaffolding questions provided at the start of each lesson. Use error analysis and Watch Out tips suggested in the textbook. Use Tip for New Teachers suggested in the textbook.
12 GEOMETRY HONORS 12 Unit Title: Chapter 2 Reasoning and Proof Targeted Standards: G.CO Prove geometric theorems. G.MG Apply geometric concepts in modeling situations. Unit Objectives/Conceptual Understandings: Students will be able to: Use inductive reasoning to formulate a conjecture. Use logical reasoning to prove statements are true and find counterexamples to disprove statements that are false. Determine the validity of a conditional statement, its converse, inverse, and contrapostive. Use deductive reasoning to prove a statement. Essential Questions: Why is it important to be able to think logically? Unit : Teachergenerated assessments will be used. G.CO.9 Prove theorems about lines and angles. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Unit vocabulary including: a. inductive reasoning b. conjecture c. counterexample d. negation e. ifthen statement f. hypothesis g. conclusion h. converse i. inverse j. postulate k. proof l. theorem Use logic to find counterexamples and use inductive reasoning to formulate a conjecture. Make conjectures based on inductive reasoning. Find counterexamples. Determine truth values of negations, conjunctions, and disjunctions and represent them using Venn diagrams. Analyze statements in ifthen form. Write converses, inverses, and contrapositives. Use the Law of Detachment. Organize students into small groups. Each student should come up with at least two statements that are not always true and the other students in the group should find the counterexamples. If students have difficulty understanding the truth value of conditional statements, then have students determine what type of conditional statements are never true. Have them analyze truth tables for conditional statements and find concrete examples where the hypothesis is always true and the conclusion is always false. Use formative assessments suggested in the textbook. Use Guided Practice, Check Your Understanding, HOT problems, and Spiral Reviews as needed. Use Ticket Out the Door Section 2.1 and 2.8. Use MidChapter Quiz Lesson p 135.
13 GEOMETRY HONORS 13 Unit Title: Chapter 2 Reasoning and Proof (cont.) Develop an awareness of a mathematical system, connecting definitions, postulates, and logical reasoning. Use logical reasoning to prove statements are true. Use the Law of Syllogism. Identify and use basic postulates about points, lines, and planes. Write paragraph proofs. Provide students with algebraic and geometric proofs that are missing the justifications for each step. At least one proof should contain errors. Have students fill in the justifications and explain the errors. Use SelfCheck quizzes as needed. Analyze statements in ifthen form. Write the converse, inverse, and contrapositive of ifthen statements. Learn to use the Law of Detachment and the Law of Syllogism. Use deductive reasoning to prove a statement. Construct and justify statements about geometric figures, using basic postulates and paragraph proofs. Use algebra to write twocolumn proofs. Use properties of equality to write geometric proofs. Write proofs involving segment addition. Write proofs involving congruence. Write proofs involving supplementary and complementary angles. Use algebra to write twocolumn proofs and use the properties of equality to write geometric proofs. Write proofs involving congruent and right angles. Write proofs involving segment addition and segment congruence.
14 GEOMETRY HONORS 14 Unit Title: Chapter 2 Reasoning and Proof (cont.) Write proofs involving supplementary and complementary angles. Write proofs involving congruent and right angles. Use deductive reasoning to prove a statement. Resources: Leveled Worksheets, Personal Tutor, Virtual Manipulatives, 5Minute Checks found on connect.ed.mcgrawhill.com Geometer Sketchpad Graphing Calculator Teaching Geometry with Manipulatives Instructional Adjustments: Use Leveled Worksheets. Use Lesson Resources to provide differentiation. Use scaffolding questions provided at the start of each lesson. Use error analysis and Watch Out tips suggested in the textbook. Use Tip for New Teachers suggested in the textbook.
15 GEOMETRY HONORS 15 Unit Title: Chapter 3 Parallel and Perpendicular Lines Targeted Standards: G.CO Experiment with transformations in the plane. G.MG Apply geometric concepts in modeling situations. G.GPE Translate between the geometric description and the equation for a conic section. Unit Objectives/Conceptual Understandings: Students will be able to: Make conjectures about lines and determine the validity of the conjectures. Use construction to explore attributes of geometric figures to make conjectures about geometric relationships. Make conjectures about angles and determine the validity of the conjectures. Find areas of regular polygons, circles, and composite figures. Use slopes of equations of lines to investigate geometric relationships, including parallel lines and perpendicular lines. Use one and twodimensional coordinate systems to represent lines. Essential Questions: Why do we have undefined terms such as point and line? How can we use undefined terms? Unit : Teachergenerated assessments will be used. Progress Indicators G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G.CO.9 Prove theorems about lines and angles. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Unit vocabulary including: a. parallel lines b. skew lines c. parallel planes d. transversal e. interior angles f. exterior angles g. corresponding angle h. slope i. rate of change j. slopeintercept form k. pointslope form l. equidistant Identify relationships between two lines or two planes. Name angle pairs formed by parallel lines and Identify the relationship between two lines or two planes. Name angle pairs formed by parallel lines and transversals. Identify the relationships between two lines or planes. Name angle pairs formed by parallel lines and transversals. Use properties of parallel lines to determine congruent angles. Graph and write equations of lines given characteristics Use masking tape to mark two parallel lines and a transversal on the floor. Have pairs of students stand in angles that are congruent or supplementary, and have them explain whether their angles are alternate interior, alternate exterior, corresponding, or consecutive interior angles. Have students graph y = x 2 on a coordinate plane. They can use a graphing calculator to generate the graph. Explain that a tangent line intersects the graph in one place. Have them predict where the tangent line of Use formative assessments suggested in the textbook. Use Guided Practice, Check Your Understanding, HOT problems, and Spiral Reviews as needed. Use Ticket Out the Door Section 3.1 and 3.5. Use MidChapter Quiz Lesson p 197.
16 GEOMETRY HONORS 16 Progress Indicators G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Unit Title: Chapter 3 Parallel and Perpendicular Lines (cont.) transversals. Use theorems to determine the relationships between specific pairs of angles. Use algebra to find angle measurements. Find slopes of lines. Use slope to identify parallel and perpendicular lines. Write an equation of a line given information about the graph. Solve problems by writing equations. Recognize the angle relationships that occur when parallel lines are cut by a transversal. such as two points, a point and a slope, or a slope and y intercept. Recognize the angle relationships that occur when parallel lines are cut by a transversal. Use angle relationships to prove that lines are parallel. Use angle relationships to prove that two lines are parallel. Use angle relationships to prove that lines are parallel. Find the distance between a point and a line. Find the distance between two parallel lines. Resources: Leveled Worksheets, Personal Tutor, Virtual Manipulatives, 5Minute Checks found on connect.ed.mcgrawhill.com Geometer Sketchpad Graphing Calculator Teaching Geometry with Manipulatives their function will be located. Have them sketch the line and predict the slope of the line. Explain to students that they will learn more about tangent lines to functions as they begin their study of calculus. Instruct students to draw two lines cut by a transversal with given specific angle measure criteria. The students may work together in small groups of 3 or 4 to discuss whether the lines must be parallel. Facilitate the discussions so that students discern that more angle measures can be found with certainty when the lines are parallel than when the lines are not parallel. Instructional Adjustments: Use SelfCheck Quizzes as needed. Use Leveled Worksheets. Use Lesson Resources to provide differentiation. Use scaffolding questions provided at the start of each lesson. Use error analysis and Watch Out tips suggested in the textbook. Use Tip for New Teachers suggested in the textbook.
17 GEOMETRY HONORS 17 Unit Title: Chapter 4 Congruent Triangles Targeted Standards: G.CO Experiment with transformations in the plane. G.SRT Understand similarity in terms of similarity transformations G.GPE Translate between the geometric description and the equation for a conic section. Unit Objectives/Conceptual Understandings: Make conjectures about polygons. Use numeric and geometric patterns to make generalizations about geometric properties. Use logical reasoning to prove statements are true. Essential Questions: How can you compare two objects? How can you tell if two objects are congruent? How can you tell if two triangles are congruent? Unit : Teachergenerated assessments will be used. G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are Unit vocabulary including: a. equiangular triangle b. equilateral triangle c. isosceles triangle d. scalene triangle e. auxiliary line f. congruent g. congruent polygons h. corresponding parts i. included angle j. included side k. base angle l. transformation m. preimage n. image o. reflection Identify and classify triangles by angle measures and side measures. Apply the Triangle AngleSum Theorem and the Exterior Angle Theorem. Name and use corresponding parts of congruent triangles. Prove triangles congruent using the definition of congruence Interpersonal Students work in groups of 2 or 3 to explore the triangle classifications. Ask students to explore and discuss questions such as these: Can you draw an equiangular triangle with a 90 angle? Can you draw a right triangle that has an obtuse angle? Facilitate the discussions so that students discover which triangle classifications are mutually exclusive and which are not Suggested quiz after section 4.4 Exit tickets provided in text Teachergenerated exit tickets Suggested quiz after section 4.6 Teachergenerated formative/summative assessment
18 GEOMETRY HONORS 18 Unit Title: Chapter 4 Congruent Triangles (cont.) congruent. G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G.CO.10 Prove theorems about triangles. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. p. translation q. rotation Triangle Angle Sum Theorem Properties of Triangle Congruence Triangle Congruence Postulates: SSS, SAS, ASA, AAS, HL Isosceles Triangle Theorem Coordinate Proof Use the SSS and SAS Postulates to test for triangle congruence. Use the ASA and AAS Postulates to test for triangle congruence. Use properties of isosceles and equilateral triangles. Identify reflections, translations, and rotations. Verify congruence after a congruence transformation. Position and label triangles for use in coordinate proofs. Write coordinate proofs. Multiple Representations In Exercise 45, students investigate the sum of the measures of the exterior angles of a triangle using geometric sketches, a table, a verbal description, and a paragraph proof. Extension Have students draw ΔABC with vertices A( 8, 8), B( 2, 5), and C( 8, 2). Next, have students draw ΔPTS with vertices P(8, 8), T(2, 5), and S(8, 2). Ask students how they can verify that the corresponding sides of the triangles are congruent. Further, facilitate discussion about whether the corresponding angles of ΔABC and ΔPTS are congruent. Students can use the distance formula to prove that corresponding sides are congruent. Further discussion about the angles may include suggestions that the triangles are exactly the same because one is a reflection of the other, or that the equal side lengths require equal angles.
19 GEOMETRY HONORS 19 Unit Title: Chapter 4 Congruent Triangles (cont.) Extension Rotations, reflections, and translations are used to create many works of art. Have students explore using these transformations to create patterns. Students should begin with a single figure in the coordinate plane and use various transformations to turn their figure into an artful pattern. Students should record each pattern used so that the design can be repeated. Resources: Essential Materials, Supplementary Materials, Links to Best Practices, Leveled Worksheets, Personal Tutor, Virtual Manipulatives, 5Minute Checks, Graphing Calculator Instructional Adjustments: Leveled Worksheets, Differentiation Resources, Scaffolding Questions, Use Watch Out questions provided in the text
20 GEOMETRY HONORS 20 Unit Title: Chapter 5 Relationships in Triangles Targeted Standards: G.CO Prove theorems about triangles and apply geometric methods to solve problems Unit Objectives/Conceptual Understandings: Students will be able to identify and use perpendicular bisectors, angle bisectors, medians and altitudes in triangles, and recognize and apply properties of inequalities to angles and sides of triangles. Essential Questions: What information do we get from knowing that a segment is a median, altitude or angle bisector of a triangle? What inequalities apply to the relationships of angles and sides in triangles? Unit : Teachergenerated assessments will be used. G.CO.10 Prove theorems about triangles G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios) Unit vocabulary including: a. perpendicular bisector b. point of concurrency c. circumcenter d. incenter e. median f. centroid g. altitude h. orthocenter Exterior Angle Inequality AngleSide Inequalities Construct viable arguments Use perpendicular bisectors in triangles Use angle bisectors in triangles Use medians in triangles Use altitudes in triangles Write indirect proofs Use RealWorld Example 2 on Pg 326 to connect the Circumcenter Theorem to reallife problems Use Interior Design Example on Pg 347 to connect AngleSide Relationships to the real world Use the Cabri Jr. application on a TI83/84 Plus graphing calculator to discover properties of triangles as outlined on Pg 363 Suggest quizzes after Sec 5.2 and 5.5 Use Exit Ticket included in Sec 5.2 Selected exercises can be used as Formative s as suggested in the textbook The Triangle Inequality Hinge Theorem Use The Triangle Inequality to identify possible triangles Use the Hinge Theorem to make comparisons in two triangles
21 GEOMETRY HONORS 21 Unit Title: Chapter 5 Relationships in Triangles (cont.) Resources: Leveled Worksheets, Personal Tutor, Virtual Manipulatives, 5Minute Checks Use graphing calculator Teaching Geometry with Manipulatives Instructional Adjustments: Use Leveled Worksheets Use Lesson Resources to provide differentiation Use scaffolding questions provided at the start of each lesson
22 GEOMETRY HONORS 22 Unit Title: Chapter 6 Quadrilaterals Targeted Standards: G.CO and G.MG Prove theorems about parallelograms, apply geometric methods to solve problems, use coordinates to prove geometric theorems Unit Objectives/Conceptual Understandings: Students will recognize properties of quadrilaterals that determine the most descriptive quadrilateral and apply properties of polygons to solve problems Essential Questions: What properties determine a parallelogram and/or special parallelograms? Unit : Teachergenerated assessments will be used G.MG.1 Use geometric shapes, their measures and their properties to describe objects G.CO.11 Prove theorems about parallelograms G.GPE.4 Use coordinates to prove simple geometric theorems algebraically G.MG.3 Apply geometric methods to solve problems Unit Vocabulary including: a. diagonal b. parallelogram c. rectangle d. rhombus e. square f. trapezoid g. bases h. legs of a trapezoid i. base angles j. isosceles trapezoid k. midsegment of a trapezoid l. kite Verify that a quadrilateral is a parallelogram Prove that a set of points forms a parallelogram in the coordinate plane Determine whether a parallelogram or a quadrilateral is a rectangle, square or rhombus Apply properties of kites and trapezoids Consider using the Graphing Technology Lab on Pg 412 Assign the Modeling problem (#32) on Pg 427 Consider giving quizzes after Section 6.1, 6.3 and 6.6 Selected exercises can be used as Formative s as suggested in the textbook Properties of: a. parallelogram b. rectangle c. rhombus d. square e. trapezoid
23 GEOMETRY HONORS 23 Unit Title: Chapter 6 Quadrilaterals (cont.) f. isosceles trapezoid g. kite How to find the sum of the measures of the interior and exterior angles of a polygon Resources: Leveled Worksheets, Virtual Manipulatives, Animations, Personal Tutor Graphing calculators Teaching Geometry with Manipulatives Instructional Adjustments: Use Leveled Worksheets Use Lesson Resources to provide differentiation Use scaffolding questions provided at the start of each lesson
24 GEOMETRY HONORS 24 Unit Title: Chapter 7 Proportions and Similarity Targeted Standards: G.SRT Understand similarity, define trigonometric ratios, and apply trigonometry in problem solving. Unit Objectives/Conceptual Understandings: Students will be able to identify similar triangles. Essential Questions: How do we prove that triangles are similar? How can we use proportional relationships in similar triangles to find missing dimensions? Unit : Teachergenerated assessments will be used. G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost, working with typographic grid systems based on ratios) G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar, explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G.SRT.4 Prove theorems about triangles G.SRT.5 Use congruence and similarity criteria for triangles to Unit vocabulary including: a. ratio b. proportion c. cross products d. similar polygons e. similar ratio f. scale factor g. midsegment of a triangle h. fractal i. iteration j. selfsimilar k. dilation l. similarity transformation m. scale factor of a dilation n. scale model o. scale drawing p. scale How to identify similar triangles Prove that triangles are similar Use proportional relationships of corresponding segments of similar triangles to solve problems Use scale factors to solve problems. Consider doing the Graphing Technology Lab which relates the Fibonacci Sequence and Ratios (Pg 468) The Geometry Lab on Pg 488 uses similar triangles to prove the slope criteria for perpendicular and parallel lines The Geometry Lab on Pg investigates iteration and fractals Quizzes are suggested after Section 7.2, 7.4 and 7.6 Selected exercises can be used as Formative s as suggested in the textbook
25 GEOMETRY HONORS 25 Unit Title: Chapter 7 Proportions and Similarity (cont.) solve problems and to prove relationships in geometric figures. How to identify similarity transformations Resources: Leveled Worksheets, Virtual Manipulatives, Animations, Personal Tutor Graphing calculators to be used with the Lab on Pg 468 Teaching Geometry with Manipulatives Instructional Adjustments: Use Leveled Worksheets Use Lesson Resources to provide differentiation Use scaffolding questions provided at the start of each lesson
26 GEOMETRY HONORS 26 Unit Title: Chapter 8 Right Triangles and Trigonometry Targeted Standards: G.SRT Solve problems involving right triangles and apply trigonometry to right triangles Unit Objectives/Conceptual Understandings: Students will be able to solve problems involving right triangles, using Pythagorean Theorem, special right triangles, and trigonometric ratios Essential Questions: How do we use the Pythagorean Theorem and trigonometric ratios to solve problems Unit : Teachergenerated assessments will be used. G.SRT.4 Prove theorems about triangles G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures G.SRT.6 Understand that by similarity, the side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems Unit vocabulary including: a. Geometric mean b. Pythagorean triple c. Trigonometry d. Trigonometric ratio e. Sine f. Cosine g. Tangent h. Inverse sine i. Inverse cosine j. Inverse tangent k. Cosecant l. Secant m. Cotangent n. Angle of elevation o. Angle of depression p. Law of Sines q. Law of Cosines r. Vector s. Magnitude t. Direction u. Standard position v. Component form w. Resultant Find the geometric mean between two numbers Solve problems involving relationships between parts of a right triangle and the altitude to the hypotenuse Solve problems using the Pythagorean Theorem and its Converse Use properties of and triangles to solve problems Find trigonometric ratios using right triangles Use trigonometric ratios to find angle measures in right triangles Solve problems involving angles of elevation and depression Use the Cabri Jr. application on a graphing calculator to explore trigonometry, the study of the patterns in right triangles Make real life connections by solving problems involving playground (Pg #44), sports (Pg #11), roller coasters (Pg #35), and angles of elevation and depression Quizzes are provided in Sections 82, 84, 86, and 87. Selected exercises can be used as Formative s as suggested in the textbook Use the Exit Ticket suggested in Lesson 85 on Pg 587
27 GEOMETRY HONORS 27 Unit Title: Chapter 8 Right Triangles and Trigonometry (cont.) G.SRT.9 Derive the formula A = ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. The Pythagorean Theorem and its Converse Common Pythagorean triples Use the Law of Sines and the Law of Cosines Perform vector operations G.SRT.10 Prove the Laws of Sines and Cosines The side relationships of and triangles G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost, working with typographic grid systems based on ratios). G.GPE.6  Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Resources: Leveled Worksheets, Virtual Manipulatives, Animations, Personal Tutor Graphing calculators to be used with the Lab on Pg 567 Teaching Geometry with Manipulatives Instructional Adjustments: Use Leveled Worksheets Use Lesson Resources to provide differentiation Use scaffolding questions provided at the start of each lesson
28 GEOMETRY HONORS 28 Unit Title: Chapter 9 Transformations and Symmetry Targeted Standards: G.CO Experiment with transformations in the plane. Understand congruence in terms of rigid motions. Prove geometric theorems. G.MD Explain volume formulas and use them to solve problems. G.SRT Understand similarity in terms of similarity transformations. Unit Objectives/Conceptual Understandings: Students will be able to: Use congruence transformations to make conjectures and justify properties of geometric figures. Essential Questions: Where can transformations be found? Why is symmetry desirable? Unit : Teachergenerated assessments will be used. Progress Indicators G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular Unit vocabulary including: a. line of reflection b. center of rotation c. angle of rotation d. composition of transformations e. symmetry f. line symmetry g. line of symmetry Draw a reflection in a line of reflection and in the coordinate plane. Draw translations in a plane and in a coordinate plane. Draw reflections in a line and in a coordinate plane. Draw translations in a plane and in the coordinate plane. Verify reflections and translations as congruence Draw reflections. Draw reflections in the coordinate plane. Draw translations. Draw translations in the coordinate plane. Draw rotations. Draw rotations in the coordinate plane. Explore the effects of performing multiple transformations on a figure. Identify line and rotational symmetries in twodimensional figures. Identify plane and axis symmetries in threedimensional figures. Create three or four large coordinate grids using poster board. Provide several laminated shapes, such as rectangles, hexagons, pentagons, and trapezoids. Students can practice physically translating shapes on the grids. Students can use examples of translations in the lesson or create their own. There are many examples of objects in nature that have symmetry. Have students draw or gather examples of objects in nature for each of the types of symmetry discussed in this lesson. Ask students to list the steps in constructing a dilation. Students should include examples that illustrate the steps and list the Use formative assessments suggested in the textbook. Use Guided Practice, Check Your Understanding, HOT problems, and Spiral Reviews as needed. Use Ticket Out the Door Section 9.1 and 9.6. Use MidChapter Quiz Lesson p 649. Use SelfCheck Quizzes as needed.
29 GEOMETRY HONORS 29 Unit Title: Chapter 9 Transformations and Symmetry (cont.) Progress Indicators lines, parallel lines, and line segments. G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. transformations. Draw reflections and rotations of figures. Draw dilations. Draw dilations in the coordinate plane. properties of a dilation. G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). G.MD.4 Identify the shapes of twodimensional crosssections of threedimensional objects, and identify threedimensional objects generated by rotations of two
30 GEOMETRY HONORS 30 Unit Title: Chapter 9 Transformations and Symmetry (cont.) Progress Indicators dimensional objects. G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor. Resources: Leveled Worksheets, Personal Tutor, Virtual Manipulatives, 5Minute Checks found on connect.ed.mcgrawhill.com Geometer Sketchpad Graphing Calculator Teaching Geometry with Manipulatives Instructional Adjustments: Use Leveled Worksheets. Use Lesson Resources to provide differentiation. Use scaffolding questions provided at the start of each lesson. Use error analysis and Watch Out tips suggested in the textbook. Use Tip for New Teachers suggested in the textbook.
31 GEOMETRY HONORS 31 Unit Title: Chapter 10 Circles Targeted Standards: G.CO Experiment with transformations in the plane. G.C Understand and apply theorems about circles. G.MG Apply geometric concepts in modeling situations. G.GPE Translate between the geometric description and the equation for a conic section. Unit Objectives/Conceptual Understandings: Find areas of sectors and arc lengths of circles using proportional reasoning. Use numeric and geometric patterns to make generalizations about geometric properties including properties of angle relationships in circles. Essential Questions: How can circles be used? Unit : Teachergenerated assessments will be used. G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Unit vocabulary including: a. circle b. center c. radius d. chord e. diameter f. circumference g. pi (π) h. inscribed i. circumscribed j. central angle k. arc l. tangent m. secant n. chord segment Radiuschord relationships Arcchordcentral angle relationships What students will be Identify and use parts of circles. Solve problems involving the circumference of a circle. Identify and measure central angles, arcs, and semicircles. Find arc length. Recognize and use relationships between arcs and chords. Recognize and use relationships between arcs, chords, and diameters. Find measures of Visual/Spatial Learners Instruct students to use a piece of string to estimate the circumference of discs or cylinders. Then have students measure the diameter of the object. Review the formulas for finding circumference by using the diameter and the radius. Have students find the circumference mathematically, using the diameter and then the radius. Have students compare their calculations with the estimate they found by using the string. Verbal/Linguistic Learners Have students construct a circle with two congruent Suggested quiz after section 10.2 Suggested quiz after section 10.4 Suggested quiz after section 10.6 Suggested quiz after section 10.8 Exit tickets provided in text Teachergenerated exit tickets Teachergenerated formative/summative assessment
32 GEOMETRY HONORS 32 Unit Title: Chapter 10 Circles (cont.) G.C.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. G.C.2 Derive the equation of a parabola given a focus and directrix. G.C.3 Use coordinates to prove simple geometric theorems algebraically. G.C.4 Use coordinates to prove simple geometric theorems algebraically. G.C.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Inscribed Angle Theorem Tangentradius relationships Theorems about angles formed by secants, tangents, and chords Equation of a circle in standard form What students will be inscribed angles. Find measures of angles of inscribed polygons. Use properties of tangents. Solve problems involving circumscribed polygons. Find measures of angles formed by lines intersecting on or inside a circle. Find measures of angles formed by lines intersecting outside a circle. Find measures of segments that intersect in the interior of a circle. Find measures of segments that intersect in the exterior of a circle. Write the equation of a circle. Graph a circle on the coordinate plane. chords and the perpendicular bisectors. Then have them write a proof supporting the congruency of their construction. Extension Have students describe the difference between a central angle and an inscribed angle, and how their measures are related if they intercept the same arc. Extension Have each student find another reallife example of segments of circles in construction, nature, and so on. Instruct students to write a brief explanation of the example they found, construct a sketch of the example, insert accurate measurements if possible, and then write an equation on the information.
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